18th Mediterranean Conference on Control & Automation Congress Palace Hotel, Marrakech, Morocco June 23-25, 2010
Model-Based Control using a Lifting Approach. Eloy Garcia and Panos J. Antsaklis, Fellow, IEEE
Abstract—In this paper, discrete-time Model-Based Networked Control Systems (MB-NCS) are studied. A lifting process is applied to a general MB-NCS configuration in which the controller is connected to the actuator and plant by means of a digital communication network resulting in a multirate system; the controller is updated every nT time units while the plant is updated every mT time units. Here, necessary and sufficient conditions for asymptotic stability are derived in terms of the parameters n and m. The lifting process is also applied to the MB-NCS configuration when only sensor data is sent over the network. In both cases a Linear Time-Invariant (LTI) system is obtained after applying lifting techniques. Necessary and sufficient conditions are given for the asymptotic stability of the system with instantaneous feedback in terms of h, the periodic update constant, and in terms of h and for the intermittent feedback case, where is the time interval in which the loop remains closed.
I. INTRODUCTION
I
N recent years, control networks have been replacing traditional point-to-point wired systems. In networked control systems, the different elements, plants, controllers, sensors, and actuators are connected through a digital communication network with limited bandwidth. The new challenges that this implementation has brought are well documented [1]-[4]. Perhaps the most relevant is the limitation on bandwidth; many researchers have studied different problems related to bandwidth restrictions such the state estimation problem under limited network capacity [5] or the minimum bit rate required to stabilize a Network Control System (NCS) [6], [7]. Other authors have focused on reducing network communication maintaining the system stable or keeping some level of performance. Georgiev and Tilbury [8] use the packet structure more efficiently, that is, reduction on communication is obtained by sending packets of information using all data bits available; for the sequence of sensor data received, the controller needs to find a control sequence instead of a single control value. Otanez et al. [9] use deadbands at each node to record the last value sent to the network and compares that value to the current one, making a decision on sending the current information or not. Walsh, et al. [10] introduced a network control protocol TryOnce-Discard (TOD) to allocate network resources to the different nodes in a Networked Control System.
Both authors are with the Department of Electrical Engineering, University of Notre Dame, Notre Dame, IN 46556 USA (e-mail:
[email protected]). The support of the National Science Foundation under Grant No. CCF0819865 is gratefully acknowledged.
978-1-4244-8092-0/10/$26.00 ©2010 IEEE
A type of NCS called Model Based Networked Control Systems (MB-NCS) aims to reduce communication over the network by incorporating an explicit model of the system to be controlled. The state of this model is used for control when no feedback is available (open loop). When the loop is closed, the state of the model is updated with new information, namely, the state of the real system. The MBNCS framework is able to reduce network communication; consequently, the network is available for other uses, reducing time delays and bandwidth limitations. Work in MB-NCS by Montestruque and Antsaklis [11], [12] provided necessary and sufficient conditions for stability for the case when the update intervals are constant; the output feedback and network delay case were also studied. In an extension, the same authors [13] also presented results when the update intervals are time-varying and follow different probabilistic distributions. In a related work Estrada, et al. [14] introduced MB-NCS to intermittent feedback control resulting in improved performance and longer permissible update intervals. Recently, the intermittent control concept has been successfully applied to control systems [15]-[18]. In all the above work on MB-NCS it is assumed that the network exists only between the sensor and the controller node while the controller is connected directly to the actuator and plant, that is the input generated by the controller is available to the plant at all times without delays or losses. The work presented in this paper has strong connections with multirate systems [22]-[26]. Such systems arise mainly due to the limitation in sensing some variables fast enough, while the control variables can be adjusted faster. The main difference in this paper with respect to the control strategies used in multirate systems is the implementation of an explicit model to generate estimates of the state between sampling times. Note that, [24] offers a similar implementation, aiming to compare certain characteristics against a fast single rate control system. By contrast, we aim to find the specific rates that result in a stable multirate system. In this paper lifting techniques are used to derive necessary and sufficient conditions for stability when there is also a communication network between the controller and the plant, a more general and flexible implementation that uses the network in both sides of the control loop. In addition, lifting techniques are also used to derive necessary and sufficient conditions for stability for traditional MBNCS configurations, thus verifying existing results.
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II. PROBLEM STATEMENT MB-NCS make use of an explicit model of the plant which is added to the controller node to compute the control input based on the state of the model rather than on the plant state. Fig. 1 shows a basic MB-NCS configuration, where the network exists only on the sensor-controller side while the controller is connected directly to the actuator and the plant.
set up and they are related by hs=h/r, where r is some positive integer. For a discrete-time signal v(k) referred to the sub-period h/r, that is, v(0) occurs at time t=0, v(1) at t=h/r, v(2) at t=2h/r and so on, the lifted signal v is defined as follows: If v v(0), v(1), v(2),...
v
then
v(0) v(r ) v(1) v(r 1) , ,... : : v(r 1) v(2r 1)
The dimension of the lifted signal v (k ) is r times the dimension of the original signal v(k) and is regarded to the base period h, i.e. v (k ) occurs at time t=kh. For a detailed treatment on lifting signals and systems the reader is referred to [19]. See also [21], [27], and [28]. Fig. 1. Representation of a Model-Based Networked Control System.
III. INSTANTANEOUS AND INTERMITTENT FEEDBACK
The dynamics of the plant and the model are given respectively by: x(k 1) Ax(k ) Bu (k ) ˆ ˆ (k ) Bu ˆ (k ) xˆ (k 1) Ax
(1)
The MB-NCS of Fig. 1 can be seen as the linear timevarying system shown in part a) of Fig. 2, by considering an output y that is equal to xˆ when the loop is open and equal to x when we have an update (closed loop). The system after applying lifting is represented in part b) of the same figure, and is regarded as a LTI system with higher dimension input and output.
where x is the state of the plant, xˆ is the state of the model, A, B are the state space parameters of the physical system and
Aˆ , Bˆ represent the model of the system. The input u for the case of instantaneous feedback can be expressed as:
Kx(k ) u (k ) Kxˆ(k )
k ih k ih j
(2)
for i=0,1,2,….; h is the number of samples of the real plant that the sensor must wait in order to broadcast a measurement update, therefore h is an integer; and 0
